We discuss the problem of inspecting an underwater structure, such as a submerged ship hull, with an autonomous underwater vehicle (AUV). In such scenarios, the goal is to construct an accurate 3D model of the structure and to detect any anomalies (e.g., foreign objects or deformations). We propose a method for constructing 3D meshes from sonar-derived point clouds that provides watertight surfaces, and we introduce uncertainty modeling through non-parametric Bayesian regression. Uncertainty modeling provides novel cost functions for planning the path of the AUV to minimize a metric of inspection performance. We draw connections between the resulting cost functions and submodular optimization, which provides insight into the formal properties of active perception problems. In addition, we present experimental trials that utilize profiling sonar data from ship hull inspection.